The system of equations x^2+2 y^2+3 z^2=6,2 x^2+4 y^2+z^2=17,3 x^2+2 y^2+9 z^2=2, over the set o

The system of equations x^2+2 y^2+3 z^2=6,2 x^2+4...

Key Concepts

Quadratic Forms

A quadratic form is an expression involving squared terms of variables, potentially combined with cross-product terms. In the context of systems of equations, recognizing that each equation is a quadratic form helps in understanding the geometry of the solution set and the inherent restrictions that arise, such as the nonnegativity of squared terms.

Nonlinear Systems

Unlike linear systems, nonlinear systems contain equations in which variables are raised to powers other than one, such as squared terms. This classification impacts the techniques used for analysis and solution since standard linear methods do not directly apply and additional considerations like multiple possible square root values or intrinsic constraints must be taken into account.

Non-Negativity Constraints

Since any real number squared yields a nonnegative result, equations that include squared variables impose natural non-negativity constraints on the components of the system. This means that when transforming or solving quadratic systems, one must ensure that any intermediate variables defined by squares (for example, x², y², z²) satisfy this property, which can critically affect the existence of solutions.

System Consistency

Analyzing the consistency of a system, whether linear or nonlinear, involves checking if the equations are mutually compatible; that is, if they share common solutions. In systems involving quadratic forms, this often involves eliminating variables and confirming that the resulting conditions do not force impossible situations, such as requiring a nonnegative term to be negative, which indicates that the system of equations has no solution.

Transcript

00:01 So in the given question we have a system of equation that is given as x square plus 2 y square plus 3 z square is equal to 6 is equal to 6 and there is another equation that is 2 x square plus 6 plus 6 y square right so 4 y square plus z square is equal to 17 and the next equation in the system is 3x square plus 2y square plus 9 z square is equal to 2 and we are given that this system of equation over the set of real number has what kind of a solution so if this the system has a unique solution or does it have a finite number of solution or infinitely many solutions or no solutions at all? so this is what we need to find.

01:17 So what we can do over here is we can take the system of equations that is given in the question and take instead of x square let's take capital x as the variable instead of y square let's take capital y and instead of z square let's take capital z as the variable so once we take this as the variable we can rewrite the equation as x plus 2y plus 3 z plus 3 z is equal to 6 2x 2x plus 4y plus z is equal to 17 3x plus 2y plus 9 z is equal to 2.

02:12 So this is the system of equations we can form now.

02:17 Right? so in the next step what we are going to do is to solve the system of equation.

02:22 We are going to use the kramer's rule, right? the kramer's rule.

02:28 So what is the kramer's rule that we need to discuss now? so consider a system, let's in order to explain kramer's rule, let's take a system of equation given by ax1 plus by1 plus cz1 is sorry a1x plus b1y plus c1z, a1x plus b1y plus c1z is equal to d1.

03:04 A2x plus b2y plus c2z is equal to d2 where d1 d2, a2, a2 and all are coefficients and constants, right? so a3x plus b3y plus c3 z as equal to d3.

03:25 So if we have such a system of equations according to the kramer's rule, we can take the let's take the determinant delta as the coefficient matrix the determinant of the coefficient matrix which is a1 b1 c1 a2 b2 c2 and a3 b3 c3 and if we define delta x as the determinant taking the first column out of the coefficient matrix and replacing it with the constant that is given over here that is d1, d2, d3.

04:14 So if we do this, this is called the delta x determinant c1, c2, c3.

04:26 And similarly we can take delta y and delta z.

04:32 Right so delta y would be replacing the second column so a 1 a 2 a 3 d1 d2 d3 and c1 c2 c3 and similarly we can find delta z which is a 1 d1 d1 a 2 b2 d2 d2 and a 3 3 d3 d3.

04:59 So why we do this is that if we can find these according to the kramer's rule, we can find the roots of the equation as x is equal to delta x divided by delta, y is equal to delta y divided by delta and z is equal to delta z divided by delta...

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